Thermal effects on collective modes in disordered $s$-wave superconductors

We investigate the effect of thermal fluctuations on the two-particle spectral function for a disordered $s$-wave superconductor in two dimensions, focusing on the evolution of the collective amplitude and phase modes. We find three main effects of thermal fluctuations: (a) the phase mode is softened with increasing temperature reflecting the decrease of superfluid stiffness; (b) remarkably, the non-dispersive collective amplitude modes at finite energy near $\mathbf{q}=[0,0]$ and $\mathbf{q}=[\pi ,\pi ]$ survive even in presence of thermal fluctuations in the disordered superconductor; and © the scattering of the thermally excited fermionic quasiparticles leads to low energy incoherent spectral weight that forms a strongly momentum-dependent background halo around the phase and amplitude collective modes and broadens them. Due to momentum and energy conservation constraints, this halo has a boundary which disperses linearly at low momenta and shows a strong dip near the $[\pi ,\pi ]$ point in the Brillouin zone.